MHB Why Does tan(x + pi/2) Equal -cotx in Trigonometry?

AI Thread Summary
The equation tan(x + pi/2) = -cot(x) is derived by applying the co-function identity, which states that tan(x + pi/2) can be rewritten as cot(-x). Since the cotangent function is odd, this leads to the conclusion that tan(x + pi/2) equals -cot(x). The discussion notes that using the tangent of a sum formula is not viable due to the undefined nature of tan(pi/2). Alternative methods, such as using sine and cosine addition rules or limits, are mentioned but not pursued in detail. The explanation emphasizes the relationship between tangent and cotangent through trigonometric identities.
mathdad
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I decided to review a little trigonometry.

Why does tan(x + pi/2) = -cotx?

I cannot use the tangent of a sum formula because
tan(pi/2) does not exist.

How about tan(x + pi/2) = [sin(x + pi/2)]/[cos(x + pi/2)] and then apply the addition rules for sine and cosine?
 
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You could certainly do that, however, you could also write:

$$\tan\left(x+\frac{\pi}{2}\right)=\tan\left(\frac{\pi}{2}-(-x)\right)$$

Using a co-function identity, we obtain:

$$\tan\left(x+\frac{\pi}{2}\right)=\cot\left(-x\right)$$

Using the fact that the cotangent function is odd, we have:

$$\tan\left(x+\frac{\pi}{2}\right)=-\cot\left(x\right)$$ :D

You can also use the angle-sum identity for tangent, if you then use a limit and L'Hôpital's Rule for the resulting indeterminate form, however we'll keep this pre-calc. ;)
 
You know the short cut way. Cool.
 
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